Science:Math Exam Resources/Courses/MATH152/April 2016/Question B 06 (b)
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Question B 06 (b) 

Consider the threecomponent differential equation . The matrix has real entries. It has an eigenvalue and an eigenvalue with corresponding eigenvectors and (b) Write the solution of the differential equation with initial data . Your solution must be in real form, that is is cannot involve complex numbers or complex exponentials. 
Make sure you understand the problem fully: What is the question asking you to do? Are there specific conditions or constraints that you should take note of? How will you know if your answer is correct from your work only? Can you rephrase the question in your own words in a way that makes sense to you? 
If you are stuck, check the hint below. Consider it for a while. Does it give you a new idea on how to approach the problem? If so, try it! 
Hint 

Use the general solution found in part (a) together with the given initial data to solve for the unknown constants. To obtain a solution in real form, replace the complex terms and that appear in the general solution by and . 
Checking a solution serves two purposes: helping you if, after having used the hint, you still are stuck on the problem; or if you have solved the problem and would like to check your work.

Solution 

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Please rate my easiness! It's quick and helps everyone guide their studies. First we put the general solution from part (a) into real form by replacing the complex term and its conjugate by its real and imaginary parts. Since and we then have that the general solution in real form is:
We are given the initial data Recall that and . Substituting in the general solution, we get
which gives and This implies that and So the solution of the differential equation satisfying the initial data is 